Question

18. \( \left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right) \)

Ask by Beck Bird. in South Africa
Jan 23,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The expression simplifies to \( \frac{6561 - 162x^{2} + x^{4}}{81} \).

Solution

Calculate or simplify the expression \( (3-x/3)*(3+x/3)*(3-x/3)*(3+x/3) \). Simplify the expression by following steps: - step0: Solution: \(\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\) - step1: Subtract the terms: \(\frac{9-x}{3}\times \left(3+\frac{x}{3}\right)\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\) - step2: Add the terms: \(\frac{9-x}{3}\times \frac{9+x}{3}\times \left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\) - step3: Subtract the terms: \(\frac{9-x}{3}\times \frac{9+x}{3}\times \frac{9-x}{3}\times \left(3+\frac{x}{3}\right)\) - step4: Add the terms: \(\frac{9-x}{3}\times \frac{9+x}{3}\times \frac{9-x}{3}\times \frac{9+x}{3}\) - step5: Multiply the terms: \(\frac{\left(9-x\right)\left(9+x\right)}{9}\times \frac{9-x}{3}\times \frac{9+x}{3}\) - step6: Multiply the terms: \(\frac{\left(9-x\right)^{2}\left(9+x\right)}{27}\times \frac{9+x}{3}\) - step7: Multiply the terms: \(\frac{\left(9-x\right)^{2}\left(9+x\right)\left(9+x\right)}{27\times 3}\) - step8: Multiply the terms: \(\frac{\left(9-x\right)^{2}\left(9+x\right)^{2}}{27\times 3}\) - step9: Multiply the terms: \(\frac{\left(9-x\right)^{2}\left(9+x\right)^{2}}{81}\) - step10: Simplify: \(\frac{\left(81-x^{2}\right)^{2}}{81}\) - step11: Expand the expression: \(\frac{6561-162x^{2}+x^{4}}{81}\) The expression \( \left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right) \) simplifies to \( \frac{6561-162x^{2}+x^{4}}{81} \).

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Beyond the Answer

To simplify \( \left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right)\left(3-\frac{x}{3}\right)\left(3+\frac{x}{3}\right) \), we can first recognize that \( \left(3 - \frac{x}{3}\right)\left(3 + \frac{x}{3}\right) \) resembles the difference of squares pattern. Let \( a = 3 \) and \( b = \frac{x}{3} \). So we can rewrite the expression: \[ (3 - \frac{x}{3})(3 + \frac{x}{3}) = 3^2 - \left(\frac{x}{3}\right)^2 = 9 - \frac{x^2}{9} = \frac{81 - x^2}{9} \] Now, we notice that there are two identical pairs of these products: \[ \left((3 - \frac{x}{3})(3 + \frac{x}{3})\right)^2 = \left(\frac{81 - x^2}{9}\right)^2 = \frac{(81 - x^2)^2}{81} \] Putting it all together, we find: \[ \left(3 - \frac{x}{3}\right)\left(3 + \frac{x}{3}\right)\left(3 - \frac{x}{3}\right)\left(3 + \frac{x}{3}\right) = \frac{(81 - x^2)^2}{81} \] So, the final answer is: \[ \frac{(81 - x^2)^2}{81} \]

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